A king decides that his country needs more women, so he decrees that families are to be allowed to continue to have children only if they give birth to girls. As soon as a boy is born, the family may not have anymore children. The king says, "This way we'll have families with 2 girls and a boy or 4 girls and a boy - we're sure to have a female heavy population now!" Assuming that the chances of having a girl or a boy are equal, does the king's plan succeed?

After this, I'm assuming that there's at the most one more girl born before a boy is, ending everyone's right to have children. The only reason the girls' side ends up with more is due to the fact that I rounded that way on the decimals. It could obviously be more boys, as well.

... unless, of course, you have the sad situation that exists in some parts of China, where baby girls are 'disposed of' (either through selective abortion or worse) until a boy is produced. They currently have 1.35 boys for every girl...

So, if Sum[n=1->inf](n/(2^n))>2 then there are more girls (and it isnt).

However, intuitively, since all births are independent of each other, you know that the King's plan will have no effect. It would be like picking '3' in the lottery because it 'hasn't come up much recently'. Each birth has a 50% chance of a girl no matter what the previous births have been.